GPG Thermal V6

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No. 84

Measurem ent andModel l i ng of Sel f -

Heat ing inP iezoe lec t r ic

Mat er ia ls andDevices

M STEWART & M G CAIN


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Measurement Good Practice Guide No 84

Measurement and Modelling of Self-Heating

in Piezoelectric Materials and Devices

Mark Stewart and Markys G Cain

Engineering and Process Control Division

National Physical Laboratory

Summary

Failures due to thermal issues are common in high power piezoelectric devices. This

Measurement Good Practice Guide aims to give engineers an understanding of the problems,

where they occur and how to avoid them. The guide covers ways and methods to predict the

temperature rise seen, based on simple analytical models and the use of Finite ElementModels. It reviews all the previous methodologies that people have used, and introduces

some new techniques, applying them in a series of case studies that embody most of the

conditions that are seen in real systems. The thermal property data needed for the thermal

models, such as specific heat capacity and thermal diffusivity are measured, along with a

summary of findings by other workers.


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Crown Copyright 2005

Reproduced by permission of the Controller of HMSO

ISSN 13686550

National Physical Laboratory

Teddington, Middlesex, United Kingdom, TW11 0LW

Acknowledgements

This work was supported by the DTI NMS programme Measurement for Materials

Processing and Performance, MPP, and this report forms deliverables MPP1.1/M8/D1.

Thanks are also due to T Amato, (PURAC), F Rawson (FFR Ultrasonics Ltd.) and DHazelwood (R&V Hazelwood Associates) for the loan of equipment, support and advice.

Further Information

For further information on Materials Measurementcontact the Materials Enquiry Point at the

National Physical Laboratory:

Tel: 020 8943 6701

Fax: 020 8943 7160

E-mail: [email protected]

Website: www.npl.co.uk


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Measurement and modelling of self-heating in piezoelectric

materials and devices

Contents

1 Introduction .................................................................................................................... 1

1.1 Where does self heating occur?...........................................................................2

1.2 How does it fail?..................................................................................................2

1.3 Thermal runaway.................................................................................................3

1.4 Examples .............................................................................................................3

2 How to minimise self-heating ........................................................................................5

3 Measuring the temperature of piezoelectric devices ...................................................8

4 Modelling.......................................................................................................................12

4.1 Energy balance ..................................................................................................12

4.2 Heat transfer processes......................................................................................12

4.3 Piezoelectric device operation...........................................................................13

4.3.1 Off resonant drive..............................................................................13

4.3.2 Resonant drive ...................................................................................14

4.3.3 Non CW operation, pulse drive and low duty cycle operation .........14

4.4 Modelling of heat generation.............................................................................15

4.4.1 Dielectric heating ..............................................................................15

4.4.2 Strain heating.....................................................................................16

4.4.3 Dielectric or strain heating ................................................................17

4.5 Modelling of heat transfer .................................................................................18

4.5.1 Steady state heat transfer ................................................................... 18

4.5.2 Transient heat transfer .......................................................................21

4.5.3 Temperature dependent heat generation............................................ 21

4.5.4 Finite element solutions..................................................................... 23

4.5.5 Example FEA solution ...................................................................... 25

5 Case studies ................................................................................................................. 27

5.1 Case 1: Off resonant drive of a simple monolithic ceramic disc suspendedin air.................................................................................................................... 27

5.1.1 Newtons law of cooling ................................................................... 29

5.1.2 Model assumptions.............................................................................31

5.1.3 Thermal runaway................................................................................ 32

5.1.4 Conclusion..........................................................................................34

5.2 Case 2: Resonant drive of simple monolithic ceramic suspended in air ............34

5.2.1 Experimental.......................................................................................34

5.2.2 Modelling ........................................................................................... 36

5.2.3 Conclusion..........................................................................................395.3 Case 3: Resonant drive of a clamped ceramic (bolted Langevin transducer) ....39

5.3.1 Experiment .........................................................................................39


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5.3.2 FEA modelling ...................................................................................41

5.3.3 Conclusion..........................................................................................44

6 Concluding Comments................................................................................................. 45

Appendix: Thermal property data of piezoelectric materials ...........................................46

A.1 Specific heat capacity, Cp (J/kgK ) ................................................................... 46

A.2 Thermal conductivity, k(W/mK)......................................................................48

A.2.1 Errors ..................................................................................................49

A.3 Thermal diffusivity, (m2/s)..............................................................................50

Acknowledgements ................................................................................................................52

References ...............................................................................................................................52


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1 Introduction

There are many uses of piezoelectric ceramics where the desire for increased power

output means increased drive levels, which subsequently can lead to thermal problems

within the device. Applications such as

Ultrasonic Cleaning

Ultrasonic Welding

Sonar Transducers

Diesel Injectors

Ultrasonic Sewage Treatment

all use piezoelectric materials operated at high drive levels, where thermal loading on

the device becomes an issue, and where potentially expensive cooling is needed tomaintain device performance.

When piezoelectric materials are used as actuators they make use of the indirect

piezoelectric effect, where the application of an electric field gives rise to an internal

strain. In this solid-state energy transformation there will always be a balance between

electrical energy input and work done by the device. The coupling coefficient, k, is

used to describe this efficiency for an ideal case where there are no losses. Here, k is

essentially the ratio of the open circuit compliance to the short circuit compliance. For

most real piezoelectric materials this conversion process is also associated with losses- both mechanical and dielectric. These losses manifest themselves in the form of

heat, causing a temperature rise in the device, which, depending on the thermal

boundary conditions can be detrimental to device performance. This self-heating

effect is most often encountered in resistive components and is termed Joule

Heating. However, it is also seen in non-ideal dielectric materials where the

dielectric loss gives rise to internal heat generation. To a first approximation,

piezoelectric actuators can be thought of as a non-ideal or lossy dielectric but, because

the material is moving, additional mechanical terms are needed to model this

behaviour. If the energy loss to the surroundings is greater than the internal powergeneration, then the sample will eventually reach an equilibrium temperature. If the

sample losses are greater than those to the environment, or if the losses increase with

increasing temperature, then the sample will heat up until some catastrophic event is

reached - such as the soldered connections failing, softening of adhesives, or

depolarisation of the material.

There are several factors that limit the high power operation of piezoceramics in

dynamic applications, such as sonar or ultrasonic welding transducers [1].

1. The dynamic mechanical strength of the ceramic

2. Reduction of the efficiency due to dielectric losses

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3. Reduction in efficiency due to mechanical losses

4. Depolarisation of the ceramic due to the applied electric field

5. Depolarisation of the ceramic due to temperature rise

The first can be largely overcome by correct prestressing or biasing of the ceramic, in

order to limit large tensile stresses. The fourth factor, electric field, can be ignored for

dynamic operations, since by the time this has occurred the field would be sufficient

to cause failure by factors 2, 3 or 5. The most common causes are factors 2, 3 and 5,

and the dominant factor depends on the type of operation.

1.1 Where does self heating occur?

In many cases this self heating does not present a significant problem, for instance in

quartz crystals used for timing in electronics, the materials intrinsically low loss and

the very low drive levels means that self heating does not adversely affect timing.

However, in an ultrasonic humidifying unit, if the water reservoir is allowed to

evaporate completely, they will fail due to overheating and for this reason they will

usually have a cut out to prevent operation when the water runs out. Typically most

self-heating related failures are in resonant continuous wave (CW) operation, since

these are likely to be the most aggressive in terms of power input. However, even in

non resonant operation, for instance in multilayer stacks where soft compositions are

used, self heating can still be a significant problem.

1.2 How does it fail?

The self-heating can be benign, or it can manifest itself in several ways in the

operation of a device. At one end of the scale, the changes in material and device

parameters caused by the temperature rise will mean that a carefully designed

resonant device will not resonate at the desired frequency. At the other end of the

scale, the temperature rise can be such that material itself fails through depoling, or

through thermal stresses. Quite often, it is not necessarily the temperature rise in the

ceramic that gives rise to failure, but the ancillary components that overheat and fail.

For instance, solder may soften, insulation may breakdown, thermal expansion can

lead to the release of prestress, and softening of adhesive leads to increased losses and

failure. Tokin [2] state that the main failures with multilayer piezoelectric actuators

are deterioration of insulation resistance, short-circuit, and open-circuit. Although

this need not be due to self-heating it can be a contributing factor.

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1.3 Thermal runaway

Much of the time, the phenomenon of self heating in piezoelectric materials is

governed by linear processes and the temperature rise seen in a device reaches an

equilibrium state, where the extraction of energy balances the internal energy

generation through joule heating. It is possible, through positive feedback

mechanisms, that the increase in temperature can lead to increased internal energy

generation, leading to a rapid and uncontrolled temperature rise and subsequent

failure. This phenomenon is also found in conventional capacitors [3] and is termed

thermal runaway, where the temperature dependence of properties can lead to

positive feedback. Although thermal runaway is often associated with catastrophic

failure, in PZT (Lead Zirconate Titanate) ceramic devices, failure is usually

associated with the failure of ancillary components, such as soldered connections,adhesively bonded joints, etc rather than mechanical failure of the ceramic. If the

temperature of the device approaches the Curie temperature, then the ceramic will

become depoled, no longer piezoelectrically active, resulting in a failed transducer.

1.4 Examples

Figure 1 Typical failure in a high power ultrasonic transducer.

Figure 1 shows a typical failure in a high power ultrasonic transducer, which was

probably caused by a failure in the insulating materials, as a result of the temperature

rise, leading to dielectric breakdown and the catastrophic failure. The bunching of the

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wires together will exacerbate the overheating, due to decreased convection around

the wires, and the pinning of the cables to the rear mass will also contribute to heating

through mechanical strain.

Figure 2 shows a failure in a transducer from an ultrasonic cleaning bath. Evidence of

thermal effects can be seen in the discolouration of the epoxy bonding the

piezoelectric to the aluminium base plate. The cracking of the ceramic is very

fragmented, indicating that the failure was not purely mechanical, but it is difficult to

attribute the failure to purely a thermal cause without further investigations. The

failure of high power devices involving epoxies is highlighted in 1-3 composites,

where poor thermal transport away from the active ceramic causes temperature related

failures in the epoxy resin [4].

Figure 2 Failure in a transducer from a laboratory ultrasonic cleaning bath.

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2 How to minimise self-heating

This guide is intended to help device designers to predict the effect of joule heating in

their chosen device and its impact on device performance. If self-heating is a

significant mechanism, then there are a number of ways that the problem can be

lessened, most of which are common sense, although some may be less obvious.

Minimise Power input: The internal heat generation is dependent on the power input

to the device and this can be limited by reducing the device operating conditions, such

as driving frequency [5], field [6], or duty cycle. As many of the relevant material

properties are non linear, a small decrease in power input may give a larger decrease

in the heat dissipated in the device. Berlincourt [7] has suggested that for safe and

efficient operation the dissipated power should be limited to less than 0.5W/cm3.

Maximise heat extraction: Obviously, the more energy that is removed through

cooling processes, the less likely it is that self-heating will cause failure. This can be

achieved by adding heatsinks, forced air or liquid cooling. Conduction processes are

usually most efficient at heat removal and increasing the gauge of the power leads

will increase conduction and reduce resistive losses in the cable. Sometimes, the

orientation of the device might help the convection process remove energy more

efficiently. Hu et al [8] found that by orienting a long vibrating plate vertically, ratherthan horizontally, the increase in convective heat removal, lead to almost a halving of

the temperature rise.

Maximise device size: Many thermal problems within the microelectronics industry

come from device miniaturisation, and if size restrictions can be relaxed, benefits in

terms of thermal performance can be achieved. Since the temperature rise in a device

can be controlled to some extent by the heat extraction, for a given power level a

larger device will usually have larger, conduction, convection and radiation heat

transfer paths. Uchino et al [9] suggest for temperature rise suppression, a device witha larger surface area is to be preferred, for example a tube rather than a rod. Another

approach might be to use a device designed for higher power levels, but drive it a

smaller fraction of the maximum.

Include thermal safety cut outs: A thermal cutout is advisable, however it is not

always feasible to measure the temperature of the ceramic itself if high voltages are

involved or the device is resonant. In this case the temperature of a nearby

component, or perhaps the air or coolant media can be measured, and a cutout based

on this reading can be used. A less expensive option might be to prevent operation

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under certain conditions, for example without a coolant present, or when the device

has been on for more than a certain length of time.

Reduce internal heat energy production by choice of material: As discussed in more

detail later, the major contribution to the internal heating is dielectric losses in the

material, so choice of materials with lower dielectric losses help to minimise internal

heat generation. At a simple level this can be through selecting a hard, rather than soft

PZT composition, since the hard materials have lower dielectric losses and can

withstand higher operating temperatures because of their higher Curie temperatures.

The selection is complicated by the fact that the dielectric loss is dependent on many

factors such as driving field, compressive stresses and temperature. Berlincourt et al

[7] give several comparisons of efficiency depending on criteria such as, maximum

internal loss limited to 0.5watts/cm3.kcps, and a maximum tan delta of 0.04.

Reduce Mechanical Losses in the system: Although dielectric losses are usually the

dominant source of heat generation, mechanical losses can also add to the overall heat

generating processes. As we are considering an actuator, any motion in the device,

apart from rigid body motions, has the potential to cause a stress and therefore a

mechanical loss. It is often seen where the active material joins the passive

components, and after all, this is how ultrasonic welding works. In 1-3 composite

actuators, the active ceramic is in intimate contact with a passive epoxy, which often

has a high mechanical damping coefficient, contributing to heat generation [10].

Where the electrical leads are soldered onto the active ceramic, stresses are introduced

into the solder, which again will have a high damping coefficient. Hu et al [8] have

seen increased temperatures around solder joints on actuators, although they have

attributed this to contact resistance, which can also contribute to heating. They

suggest that making electrical contact at the nodal point, traditionally used to maintain

a high Q in a device, may be counterproductive in terms of self heating, since this is

where the strain transfer will be a maximum. Heating through mechanical losses can

also be seen in the electrical contacts depending on how the wires are routed, often

one end will be rigid and the other attached to the moving actuator.

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Solder bumps

Wire

top surface

Wire

(underneath)

Solder bumps

Wire

top surface

Wire

(underneath)

Figure 3 Thermograph of self-heating in piezoceramic disc. Colour scale from blue

(coolest) to red (hottest). Sample driven at thickness mode resonance of disc at1.745MHz. Coolest regions (blue) correspond to where power cables connect, and

hottest points (red) to the two passive solder bumps.

Figure 3 shows the effects of solder connections on a disc resonating in the thickness

mode at 1.745MHz. The electrical power to the disc supplied via two soldered wires,

one on the top surface, and one on the bottom. In addition to this, two extra dummy

solder bumps that have been added to the top surface to examine the effect of these

additions have on the temperature profile. The temperature of the device is non-

uniform for several reasons. Although the sample is operating in thickness mode, non-uniform strains are still introduced through Poissons ratio effects, which should lead

to a temperature maximum at the centre of the disc. However because of the soldered

connections, this maximum is offset towards the dummy solder bumps. The coolest

parts of the device are where the power wires are connected; here the heat transfer

through conduction in the wires leads to a smaller temperature rise. The two dummy

solder bumps show increased temperature rise, probably because of the mechanical

loss associated with the material. It is interesting to note that the solder joints

associated with the power cables do not show the heating effect, since it is

overshadowed by the increased thermal conduction in the cables.

In summary the contributions to the non-uniform self-heating in this device are in

order:

1. Non-uniform strain in the soft piezoelectric material.

2. Increased heat transfer through conduction in the power cables.

3. Increased heat generation due to mechanical losses in the solder.

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3 Measuring the temperature of piezoelectricdevices

When attempting to measure the temperature of piezoelectric materials used in

actuators, there are several practical difficulties to be overcome. Firstly, the devices

often have high voltages applied, which can have implications for safe and accurate

temperature measurement. Secondly, if the device is resonant, it may not be practical

to attach contact temperature probes, since this will interfere with the resonant nature

of the device. Thirdly, as discussed previously, when a moving actuator comes into

contact with a passive (not moving) temperature probe, there is a potential for

frictional heating, giving rise to erroneous readings. These three factors point towards

using non-contact temperature measurement methods such as optical pyrometry, or

even thermally sensitive paints. Thermally sensitive paints undergo a colour change,

sometimes permanent, on reaching a certain temperature. The accuracy is limited, but

they are cheap and essentially non-contact. If greater accuracy and range is needed

and the temperature reading needs to be interfaced to data collection systems, then

infrared (IR) temperature sensors are a reasonably economical solution. Errors can

arise in these systems if the emissivity of the sample is low, or not precisely known,

and the temperature measured by the radiation thermometer will not represent the true

temperature of the sample. Also, if the sample is small, it might not be fully filling the

field-of-view of the radiation thermometer, and the temperature measured will be an

average of the sample and its surroundings. Occasionally the sample can act as a

mirror and the sensor measures the radiation of an object from behind the sensor.

Another non-contact method of determining the temperature of a piezoceramic device

is to use the temperature dependence of the material properties as a temperature

indicator. The simplest property to use is the capacitance of the device. For instance a

soft PZT-5H composition changes permittivity by 33% over the temperature range 0

to 40 C [11], and assuming this change is linear over the range, a change of 1% in

the capacitance represents roughly a degree centigrade. The measurement ofcapacitance could be realized using the device drive wiring and electronics, however

it may be difficult to achieve the required accuracy, particularly when measuring hard

materials, where the temperature dependence of capacitance is much less. M H Lente

et al [12] have shown a one to one correspondence between sample temperature and

the polarisation during fatiguing of PZT discs. As the device fatigues, the change in

polarisation causes a change in the current drawn by the device, leading to a change in

temperature. Ronkanen et al [13] have also identified the link between temperature

and current drawn by the device, and suggest that this could be used a mechanism for

compensation of output, as the change in temperature causes a change in piezoelectricproperties such as d33.

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If contact methods are preferred, then there are several work-arounds that can be used

to mitigate some of the problems discussed before. In order to mitigate the high-

voltage danger, then the thermocouple can be placed on the ground side of the device,

or a thin insulating varnish or cyanoacrylate can be used to attach the sensor. Of

course, should the insulation or piezoceramic breakdown then there is a potential for

damage to sensitive measuring electronics and also potential for hazardous voltages to

come into contact with users. For this reason, it is advisable to have the temperature

measurement electronics electrically isolated from the outside world. From the point

of view of interfering with a resonant device and also in order to minimize ultrasonic

heating effects, it is best to have very small temperature sensors that act as point

sensors. In this respect, thermocouples are readily available in thin wire gauges as

small as 0.003 (0.076 mm) diameter and smaller. Because of their low thermal mass,

these small thermocouples are also advantageous in terms of their transient response

and also their efficacy in measuring small samples reliably. Other contact mode

sensors including Resistance Temperature Detectors (RTDs), thermistors and IC

sensors, can all in principle be used, but the most commonly used sensors for these

types of measurements are thermocouples and non-contact IR sensors. It should be

pointed out that although no evidence of problems was seen in this work, there have

been reports of thermocouples giving spurious readings in materials experiencing

high power ultrasonic vibrations. Mignogna et al [14] used copper constantan

thermocouples bonded with epoxy to various resonant and non resonant bodies, and

reported that the thermocouple would often record a temperature rise of as much as

100C, yet the sample was barely warm to the touch. Experiments to find the cause of

these problems were inconclusive. However, they state the results cast doubt on all

previous measurements where thermocouples were used to measure heat generated

during high-power ultrasonic insonation of metals. It should be stressed that the

thermocouple measurements carried out in this work were used to record heating

generated in the piezoceramic, and in general, temperature rise in the metallic parts

was through heat conduction into the part, rather than internal heat generation in the

metal.

Figure 4 shows a comparison of temperature measurements on a 30mm diameter,

1mm thick, disc of soft PZT, using a type K thermocouple and an infrared camera

sensor. The thermocouple was attached near the centre of the disc using cyanoacrylate

and the camera used gave a full field image of the device, but only results near the

thermocouple position were used in the comparison. The IR system was corrected for

emissivity of the target surface by placing the sample on a hot plate and comparing

with the thermocouple readings. The calibration curve was highly non-linear, with a

4th

order polynomial used to represent the readings, but this is probably a due to thecamera sensitivity rather than the temperature dependence of the surface emissivity.

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The readings are a maximum of 2C apart and on average less than a degree apart,

which is the order of the accuracy of the two systems. The differences could be

because of the slightly different spatial positions of the sensors, (the disc was

operating in a resonant mode so the temperature was spatially dependent), or because

of the different time constants of the sensors. The fact that the two temperature traces

cross on the heating cycle, but not on the cooling cycle could again be due to sensor

time constants, or it could be due to electrical pickup. The only difficulty encountered

in this work relating to thermocouple measurements came through electrical

interference of a high power transducer running at around 1kW. In this case, when the

power was applied readings shot to over 200C but instantaneously returned to

ambient when the power was removed. The overall conclusion is, that thermocouple

and IR sensors give identical results, except when substantial electrical interference is

present, where these effects are easily identified.

20

25

30

35

40

45

50

55

0 20 40 60 8

Time (seconds)

Temper

atureC

0

Thermocouple

Infra Red Camera

heating cooling

Figure 4 Comparison of temperature measurements of a piezoceramic disc

undergoing self-heating. Thermocouple is fixed to the disc with cyanoacrylate, IR

measurements made with IR camera and temperature readings taken near

thermocouple position.

The temperature sensors discussed so far have essentially been single point sensors.

However, much information regarding the thermal performance of a piezoelectric

device can be gained by using IR cameras. Calibration of these sensors is moredifficult since the performance of each camera pixel could be different, however the

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information is usually used to detect differences point to point in an image and how

this progresses in time. In this respect these types of images can often be misleading

since the images will often contain areas with different emissivity that will be difficult

to account for. Figure 5 shows a raw (uncorrected for emissivity) thermal image of a

thin piezoelectric disc under resonant drive in the radial mode. To highlight the

emissivity issue, one half of the discs top surface has a graphite coating, whilst this

has been removed from the other half. The graphite gives an increased signal so it is

possible to see the ring pattern produced by non-uniform self-heating, whilst this

pattern is absent in the uncoated sector (uppermost). In the uncoated region the hottest

part appears to be the soldered power connection, which is similar to the centre of the

disc. In fact this is an artefact, the solder is acting as a mirror reflecting some other

part of the scene, giving a false impression of temperature. To overcome this the

sample would normally be entirely coated with graphite.

Figure 5 Greyscale Thermograph of PC 5H disc resonating in radial mode. Sample

has been covered with graphite to give uniform emissivity, but then some of this

coating has been removed to illustrate the effect of emissivity on the image. Left hand

solder connection appears to be as hot as the centre of the disc. The circular pattern

due to nodal strain heating effects is only visible on the graphite coated sector.

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4 Modelling

The aim of the modelling described here is to be able to predict the temperature rise in

any piezoelectric device, given the sample type, applied voltage, frequency, and some

environmental conditions.

4.1 Energy balance

The temperature rise in a piezoelectric is a result of a change in its internal energy,

and that internal energy can be found by an energy balance criteria, such that:-

Change in internal energy = energy generated in device energy lost to surroundings

Each of these energy transfer processes can be a function of many different processes:

Energy generated in device =f(loss, frequency, capacitance, voltage)

Energy lost to surroundings =f(conduction, convection, radiation)

In order to model piezoelectric device behaviour as far as thermal conditions isconcerned, there is a need to understand both the heat generation, and the heat

transfer. The heat transfer is a general problem that is covered by many textbooks and

software solutions. There are many practical heat transfer problems that include

internal energy generation, such as chemical reactions, nuclear radiation, resistive

heating, where the solutions are common and the exact nature of energy generation

process is not important. The mechanisms of heat generation in piezoelectrics,

although similar to those in dielectric materials, are further complicated by the

piezoelectric coupling. We will assume that the internal energy generation is the

largest unknown in these problems and that the heat transfer constants are largelyconstant over the temperature range of interest for most cases.

4.2 Heat transfer processes

The three mechanisms of heat transfer; conduction, convection and radiation can be

described by their individual rate law [15].

Conduction, the diffusion of energy by random molecular motion can be described bythe rate equation, Fouriers Law.

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dx

dTkmWqx =)/(

2" (1)

Where k is the heat transfer coefficient for conduction, and has units of W/mK, T is

temperature and qxis the heat flux in thex direction.

Convection, which is a combination of conduction and energy transfer due to mass

motion, advection, can be described by the following rate equation, sometimes known

as Newtons law of cooling.

(2))()/( 2 = TThmWq sx

Where h is the heat transfer coefficient for convection, and has units of W/m

2K.

Values for h can range from 2 for free convection in a gas, to 20 for forced

convection, to many thousands for convection with a phase change.

Radiation, the transfer of energy by electromagnetic radiation, can be described by the

following rate equation,

(3))()/( 442 sursmx TTmWq =

where m is the emissivity, and is the Stefan-Boltzmann constant (5.67x10-8

W/m2K4)

In general the most important heat transfer processes for piezoelectric devices are

conduction and convection. Since the devices are solid state the heat transfer within

the device will be through conduction, and much of the energy transfer to the

surrounding media will be through convection into air, water or other fluids.

4.3 Piezoelectric device operation

Piezoelectric devices are used in an ever-expanding range of applications that cover a

wide range of operating regimes of frequency, power levels etc. However, as far as

internal heat generation is concerned, there are three key types of driving conditions,

resonant drive, non-resonant drive and non-CW drive.

4.3.1 Off resonant drive

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In general, the temperature of the sample/ device is uniform throughout the sample.

This is because strain throughout the sample is uniform, so each part of the sample

volume is identical, apart from those that see a different heat transfer rate, such as

sample surfaces or regions next to internal electrodes. It is usually accepted that under

these conditions the dielectric losses contribute to self-heating. Several theoretical

analyses have been developed to predict the temperature rise of devices under off

resonant conditions. Most of these models assume that the thermal conductivity and

the heat generation mechanism are independent of temperature. Over the operating

temperatures of most piezoelectric devices it is likely that the thermal conductivity is

temperature independent, however the assumption that the internal heat generation is

temperature independent may lead to underestimation of internal temperature profiles.

4.3.2 Resonant drive

Here, standing waves are set up within the sample, which introduces a non-uniform

strain and thus potentially non-uniform temperature. Non-uniform temperatures have

been observed at nodal points in resonating devices [16], [9]. It is not clear if these

non-uniform temperatures play a part in thermal runaway type behaviour. For

example, a small part of the sample may become locally hotter and, through heat

transfer, will heat up the rest of the sample. Here, the question arises: does failure

come about from a small part reaching a critical temperature, or the whole sample

reaching this temperature?

The critical factor under resonant drive conditions is that the mechanical loss is

increased, and may contribute significantly to the internal heat generation, and also

that this heat generation process is spatially dependent. Little work has been done on

predicting temperature rise under resonant conditions, and the non-uniformity of

temperature profiles.

4.3.3 Non CW operation, pulse drive and low duty cycle operation

Non CW drive waveforms of piezoceramics are often used, either because it is

necessary for the particular operating pattern, or to overcome some of the overheating

problems associated with CW operation. A diesel injector valve is an example of

pulse drive, where the duty cycle is actually very small, but the operating cycle is very

high power. In contrast, a high power ultrasonic cleaning bath may be a resonant

device, but in order to restrain the temperature rise, the operation may be limited to

duty cycles of 10% or less.

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For the ultrasonic bath example, the low duty cycle operation only slightly

complicates the heat transfer problem, in that the heat generation is modulated in the

time domain only. For the diesel injector example not only is the operation non

continuous, the drive signal will likely be square wave, meaning that the internal heat

generation will have many additional components in the frequency domain.

4.4 Modelling of heat generation

4.4.1 Dielectric heating

In its simplest form a piezoelectric device can be thought of as a capacitive load,

whose energy at any point can be modelled as the energy in a capacitor. If the

capacitor is not ideal then some of this energy will be lost, dependent on the dielectricloss tangent of the material. If we assume that all this lost energy is converted to heat

in the capacitor through dielectric heating then the power dissipated can be given by

[17]

(4)2tan2 VCfPower =

where f is the frequency, Cthe capacitance of the device, Vthe applied voltage and

tan e the dielectric loss. Dielectric heating is the mechanism for heating in a domestic

microwave oven and the temperature rise of the food in an oven can be predicted

similarly using (4), coupled with the density, specific heat capacity and the thermal

boundary conditions.

Energy dissipation in piezoelectric materials is further complicated by the materials

intrinsic non-linearity when driven under high field, such that the permittivity and

dielectric loss become field dependent [18]. Coupled with this, the permittivity and

dielectric loss will also be temperature dependent [19], [20].

The dielectric heating model has been used by many workers [21], [22] as a basis for

heat generation in piezoelectrics driven at low frequency, under off resonance

conditions. Uchino [9] used a modification to this model to predict temperature rise in

multilayer piezoelectrics under off resonance drive. Uchino [9] expresses the rate of

heat generation in the multilayer q as

effufVq = (5)

where u is the loss of the sample per driving cycle per unit volume,fis the frequency,

and Veff is the effective volume of active ceramic material. The effective volume isdependent on the amount of active ceramic in a multilayer. The loss u is given by

15


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(6)eox

Eu tan20=

which is essentially a geometry independent version of the power loss in a capacitor

seen previously (4).

4.4.2 Strain heating

Under adiabatic conditions, a material undergoing a change in stress state can undergo

a temperature rise in the same way that an ideal gas does. This temperature rise as a

result of a volume change is termed the thermoelastic effectand can be described by

the following equation

)( 321

++= T

CT

p

l (7)

where l is the thermal expansion coefficient, and i are the changes in the principal

stresses. This temperature rise is fully reversible and results in cooling on expansion

and heating on contraction. In practice these changes are of the order 0.1K or less for

fully elastic conditions in most metals. When the material is no longer perfectly

elastic the material can undergo heating effects that are sometimes described asthermoplastic heating, and are attributed variously to grain boundary motion,

dislocation movements, bond rotation in polymers and others. Here we will use the

term, strain heating to describe the process of temperature rise due to mechanical

motion in piezoelectric materials.

Under resonance drive conditions the dominant mechanism for heat generation is

thought to be due to strain heating, rather than dielectric heating. This phenomenon is

seen in many materials undergoing high power ultrasonic vibrations [14], and is

mechanically analogous to dielectric heating. In dielectric heating there is a lagbetween the applied voltage and current, which corresponds to an energy loss, which

is converted internally into heat. In strain heating there is a lag between stress and

strain, and the corresponding energy loss is converted into heat. For dielectric heating

the energy loss is proportional to the dielectric tan e times the square of the field,

whereas for strain heating, the energy loss is proportional to the mechanical loss, tan

m, times the square of the strain.

Ando [23] has used the following relationship to determine the mechanical loss for a

given volume;

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2

2S

q vv

= (8)

where v is the damping coefficient, and S the magnitude of the vibratory strain.

Other workers have used similar expressions using the applied stress, to describe the

losses due to mechanical vibrations, [9], [24], [25]. Uchino [9] determines the

hysteresis loss in a full cycle using the following expression;

(9)mE

m Xsw tan2

0=

where sE

is the compliance and X0 is the amplitude of the stress. Blotmann et al [25]

use a very similar expression, but divided by 2, to define the mechanical dissipation,where the discrepancies with Uchino are due to imprecise definitions of the cycle. Lu

and Hanagud [24], use irreversible thermodynamics to develop a model for self

heating, but similarly show the strain heating is proportional to the square of the strain

multiplied by various viscous damping coefficients.

The relationship between strain and heat generation implies that, under non-uniform

strain conditions, for example at resonance or antiresonance, the heat generation will

be spatially dependent. This is the key difference between strain heating and dielectric

heating where, in the latter, the internal heat generation is assumed uniformthroughout the volume.

Sherrit [26] has shown that for a plate sample of thickness,L and areaA, withx=0 at

the centre, the power distribution as a function of distancex is given by

=

L

xPxP m

2cos2)( (10)

where Pm is the mean power level. The cosine squared variation of temperature in a

resonating piezoelectric device has been confirmed by Tashiro[16], although the form

of the temperature variation at the fundamental resonance is not very different from a

case with uniform heat generation. Several workers [27], [28] have modelled and

measured temperature profiles of piezoelectric transformers driven at resonance, and

shown that hot spots coincide with regions of high strain.

4.4.3 Dielectric or strain heating

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In a capacitor the only internal heating mechanism is dielectric heating. However, for

piezoelectric material whenever an applied field leads to dielectric heating there will

also be some associated strain heating. In general, most workers have assumed that

under off resonance conditions dielectric heating is dominant, and ignored any strain

heating contribution. Under non-resonant operation the induced strain will most

probably be uniform throughout the sample, and because of this the errors associated

with this will be equivalent to an error in the tan e. Conversely, under resonant

conditions the dominant mechanism is assumed to be mechanical heating, and

dielectric heating is ignored [16],[26]. However, as before, where there is an applied

voltage, there will be associated dielectric heating, but this will be spatially

independent. Hu [27] has derived an expression for the internal loss per unit volume

in a piezoelectric transformer as;

(11)eax

pkxeAxp += 20 cos)(

where the first term is the spatially dependent mechanical dissipation, and the second

pe is the dielectric heating contribution. As can be seen, neglecting this contribution

will result in an offset in the temperature rise predicted, depending on the relative

contribution of this second term. Determining the mechanical and dielectric loss for a

given situation is difficult, and there are various approaches. Hu [27], uses the phase

difference between the input voltage and current to determine the overall level of

energy loss, and uses a value of 7:3 for the ratio of mechanical to dielectric loss,based on previous experimental evidence. Other workers [24], [23] use the low field

values of mechanical and dielectric loss and equations (6) and (9) to determine the

individual contributions in FEA simulations.

4.5 Modelling of heat transfer

4.5.1 Steady state heat transfer

In the preceding discussion on heat generation, the power dissipated when a certain

sinusoidal field level is applied to a device can be determined. However, this is not a

means by itself to calculate the rise in temperature of a device. To achieve this, there

must be some understanding of the materials thermal properties and the thermal

boundary conditions of the device. At the simplest level we can imagine a volume of

material that loses power P to ambient air, with a heat transfer coefficient h, (surface

conductance into air), where the heat loss is essentially dependent on the surface area

to volume ratio.

AreahTP m= (12)

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Here P is the dissipated power and m is the temperature rise of the sample.

For a simple disc shaped device the surface temperature rise is given by [29]

142

+=

dth

PT vm (13)

where dand tare the diameter and thickness of the device, and Pv is the dissipated

power per unit volume.

Equation (13) describes the surface temperature rise of a disc that has a uniform

power generation Pv, and is losing energy to the surroundings via convection only.

This assumes the temperature of the whole surface is the same, and the internaltemperature is constant, i.e. the volume is small enough for internal heat conduction

to be ignored.

The internal temperature, and its

spatial variation, of a plate,

thicknessL and areaA, where the

external temperatures are

maintained at T0, with a uniform

internal power generation, P, isgiven by

Surface

Temperature

T0

+L/2

T(x)

x

-L/2

Surface

Temperature

T0)14(Tx

4

L

kAL2

P)x(T 0

22

+

=

The external temperature T0 can

be determined by defining the

control volume and the

mechanisms of heat transfer in

the media. However, to a first

approximation T0 is equal to the

ambient temperature.

Equation (14) is valid for uniform power generation, and so would apply to off

resonance drive, with dielectric heating. Sherritt et al [26] have shown that if the

power generation in the slab is not uniform, but proportional to the square of the

strain, then the temperature distribution is given by:

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( )02

2

2

2

4

1cos

2)( T

L

xLx

kA

PLxT +

+=

(15)

This distribution can be used to predict the temperature of a device driven atresonance, and so measurements of temperature distribution of resonating devices

should be able to discriminate between dielectric heating and strain dependent

heating. Figure 6 shows the predicted temperature distributions for uniform and strain

squared heating, showing the different form of the behaviour. Unfortunately, if the

temperature scale of the cosine squared distribution is scaled down the behaviour is

very similar to that in the uniform heating case, which means that very accurate

experimental measurements are needed to definitively differentiate uniform and

spatially dependent heating. As will be seen later the spatial dependent effects are

more readily observed at the overtones than at the fundamental resonance of the

device.

20

25

30

35

40

45

50

55

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Distance (mm)

TemperatureC

Uniform heating

Cos squared heating

scaled down cos^2

Figure 6 Temperature profile of an infinite slab subject to internal heat generation.

Comparing uniform heat generation with strain dependent heating.

The preceding models only account for conduction in the sample. In order to account

for convection at the surfaces the following model gives the temperature rise at a

distancex from an infinite slab of thicknessL, with a thermal conductivity k, andheat

transfer coefficient for convection, h:

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( ) 02

2

42Tx

L

k

hL

h

PxT +

+= (16)

The maximum temperature is at the centre of the slab, x=0, and the minimum at thesurfacex=L/2.

4.5.2 Transient heat transfer

The models so far have covered steady state heat transfer, that is the predicted

temperatures and temperature distributions are of a system in an equilibrium state, and

it tells us nothing about the temperature change with time. In many applications, this

equilibrium state is the most important, since this defines the steady state operatingconditions. However, the knowledge of transient behaviour can be useful for low duty

cycle behaviour, or for the prediction of thermal runaway.

The analytical solutions for transient heat transfer are much more involved than the

steady state solution, and as a consequence the solutions are often numerical, based on

finite differences or finite elements.

As an example the analytical solution for the temperature distribution in a semi-

infinite solid, with initial temperature T0, with heat produced at a constant rate P, perunit time per unit volume is given by: [30]

k

Pxe

t

k

Px

t

xerf

k

Px

k

tPTtxT t

x

222),(

2

42

12

0

2

+

++=

(17)

where the surface is maintained at zero temperature, and is the thermal diffusivity.

4.5.3 Temperature dependent heat generation

In all the previously described models the thermal material parameters and the heat

generation processes have been considered temperature independent, with a

consequence that all the solutions will be stable. From experimental results, it appears

that the thermal diffusivity is relatively temperature independent and the thermal

conductivity and specific heat capacity increase approximately 20% over the common

operating temperature regime. However the heat generation process is likely to be

temperature dependent, as many of the dependent parameters such as dielectric

constant and loss are also temperature dependent. Inclusion of temperature dependent

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heat generation can lead to unstable solutions, which is a possible mechanism for

thermal runaway.

Adding temperature dependent heat generation to the models further complicates the

solution and makes the analytical solutions even more complex. The following is the

temperature profile of a slab length, l, with no flow of heat at x=0, and initial

temperature of 0K. The heat generation is defined by k(A+B.T), where k is the

thermal conductivity,A andB are constants whereB is the temperature dependent part

of the heat generation.

( ) ( )( ){ } ( )( )( )( )

= ++

+++

+

=

0222

2222222

2/1

2/1

121242/12cos4/412exp116

1cos

cos),(

n nnBllxnlktlBnAl

lB

xB

B

AtxT

(18)

In order for this solution to be stable, the temperature dependent part of the heat

generation, B, must be such that the following inequality holds,

2

2

4lB

< (19)

If the equality holds then there is a steady state solution to the transient problem,given by:

= 1

cos

cos)(

2/1

2/1

lB

xB

B

AxT (20)

0

2

4

6

8

10

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

Distance from Centre (m)

TemperatureC

Analytical B=0

Analytical B=A/10

FEA B=0

FEA B=A/10

Figure 7 Temperature dependent heat generation in an infinite slab, where B is the

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temperature dependent factor. Comparison of the analytical results (points) and a 1D

finite element simulation (lines).

The temperature profile determined using equation (18) is illustrated in figure 7,

calculated at long times, so it is equivalent to equation (20), and shows the effect ofincreasing the temperature dependent heat generation factor, B. Obviously the larger

B becomes the higher the temperature at the centre will become, the outside

temperature being held at zero in the model. To add other, more realistic, boundary

conditions to the model, the analytical solution increases in complexity, which is

where FEA solutions come into their own. Also included on the figure are FEA

solutions for the same problem, showing very similar results.

4.5.4 Finite element solutions

Finite element software such as ANSYS or PAFEC allows both transient and steady

state solutions to heat transfer solutions for a variety of boundary conditions,

including internal power generation, temperature dependent thermal conductivity, and

also some coupled field solutions, such as magneto thermal, i.e. the heat generated

through joule heating in electromagnet coils. However, to model the self-heating

problem observed in piezoelectrics using ANSYS, the temperature and

piezomechanical solution would have to be solved sequentially. The thermal model

would be treated as a system with internal power generation, and coupling betweenthe piezomechanical and thermal solutions could be achieved through an iterative

process.

Early work [31], using Finite Elements to model behaviour in an ultrasonic power

transducer, used the dielectric heating model to determine the internal power

generation, which was assumed uniform throughout the device. The dissipated power

was determined by fitting results to experimental temperature profiles, rather than

using the material dielectric loss and applied voltage to calculate the power.

Shankar and Hom [32], used dielectric heating alone to predict temperature related

phenomena in an electrostrictive PMN sonar transducer. Contrary to PZT, dielectric

loss in PMN decreases with increasing temperature [33], and so is not beset by the

thermal failures seen in PZT. The internal power generation was calculated using the

loss tangent and applied voltage, and this power distributed evenly through the

device. In order to simulate pulse drive and reduced duty cycle, this power input was

distributed temporally. For example, a 33% duty cycle was simulated by full power

on for 1 second, then off for 2 seconds.

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Ando [23] carried out a comprehensive simulation of transient thermal behaviour in

an ultrasonic transducer, using an iterative procedure as follows:

1. Determine the static stress and strain in the device.

2. Calculate resonant frequency, stress, and strain etc using room temperature

material parameters.

3. Determine the mechanical and dielectric heat dissipation.

4. Calculate heat diffusion and temperature distribution.

5. Calculate thermal stress, strain and thermal expansion.

6. Change material parameters due to new temperature distribution and repeat

cycle.

This routine was continued until the required time was reached. He accurately

predicted, not only the temperature rise of the device, but also its effect on the

resonant frequency of the transducer.

N Abboud et al [34] have used the PZFlex finite element package to deal with heat

generation in 1-3 piezocomposites. They point out that a coupled solution to the

problem is difficult due to the different characteristic timescales of the different

processes; the piezoelectric vibration at 150kHz, and the thermal conduction that

occurs over the order of seconds. Their solution to this problem takes the following

form:

1. Model the piezoelectric effect for one cycle and then calculate the power

generation.

2. Solve for the transient thermal problem to determine the temperature

distribution.

3. Apply any piezoelectric, mechanical, thermal property changes due to

temperature rise/fall. Repeat steps 1-3.

The solution is reached at 2 when the temperature dependence of the material

properties is negligible for the heating levels involved.

More recently Blottmann et al [25], have performed a full 3D FEA simulation of

model sonar transducers using ATILA. Mechanical and dielectric losses were

included in the model, and although the mechanical loss was constant and the low

field value, the dielectric loss was both field and temperature dependent. Again an

iterative routine was used to modify material parameters dependent on the

temperature rise from the internal losses.

Although fully coupled piezo-thermal solutions are not yet realised in FEA packages,

another approach to the problem could be to use thermally dependent internal heat

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generation. In the ANSYS package, by using point elements, heat generation rates can

be modelled using a polynomial of the form:

64

5321

AATATATAAQ +++=

where the constants Ai can be input as material parameters. Thus if it is possible to

model the temperature dependence of the heat generation rate of the piezoelectric, this

can be included in the thermal solution.

4.5.5 Example FEA solution

As shown in figure 7, 1D FEA simulations agree well with analytical solutions but

come into their own when the models become more complex.

Figure 8 2D steady state FEA solution for a sample with uniform heat generation.

Line profiles are the same as figure7, but the end effects are observed because the

slab is not infinite.

Figure 8 shows a simple example of a 2D steady state FEA solution for a sample with

uniform heat generation, and boundary conditions of zero degrees on the long sides,

and held at 5 degrees on the short sides. The temperature profile across the centre of

the sample is similar to that in figure 7, however the core temperature is slightly

25


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higher because of the increased temperature of the short sides. This is relatively trivial

example for FEA, however the 2D analytical solution would be much more complex.

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5 Case studies

In order to demonstrate some of the predictive modelling described in this document,

a number of case studies will be discussed, where the models are compared to

experimental measurements. Each case study is intended to highlight a different

problem faced in modelling piezoelectric thermal behaviour.

5.1 Case 1: Off resonant drive of a simple monolithicceramic disc suspended in air

Rationale: This is almost the simplest case encountered; the sample is thin, so there is

no significant temperature profile through the sample. A balance between internal

heat generation, and heat removal via convection from the surface, governs the

sample surface temperature and there is no thermal conduction. The sample is

operating off resonance, so the internal heat generation is assumed uniform

throughout the sample.

The sample is a 0.2 mm thick,

10mm diameter disc of soft PC

5H composition, mounted as

shown in figure 9. The sample

was placed in a large enclosure,

approximately 0.03 cubic

metres (1 cubic foot), to control

the air movement around the

sample. The temperature of the

sample was measured with a

thermocouple, and comparisons

were made using an IR sensor,

but little difference was seen.

The sample was driven bipolar

at frequencies of 100 Hz to 5kHz and the current and voltage measured during each

run. The data collection routine had a thermal cut-off point, should the sample surface

temperature reach 100C, in order to prevent irreversible changes in the material.

Otherwise the applied power would be automatically removed once a stable

temperature was reached.

Figure 9 PZT disc sample, suspended in air,

showing power cables (thick) wires and

thermocouple (thin wire).

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23

27

31

35

0 50 100 150 200

Time (seconds)

SurfaceTemperatureC

35V

38.5V42V

45.5V

49V

52.5V

56V

Figure 10 Example heating/cooling curves for a thin piezoelectric disc driven at

5 kHz. Power is automatically cut off when sample temperature stabilises, resulting in

cooling curves.

Figure 10 shows an example of heating/cooling curves for these experiments, for

applied voltages at 5kHz, where generally the self-heating leads to an equilibrium

temperature. The equilibrium temperature depends on a number of variables including

the geometry, convection coefficient as well as voltage and frequency, but as stated

previously is essentially a balance of the input energy against the energy removal

processes. This balance can be described by the following equation

Est (Energy Stored in Sample) = Eg(Internal Energy Generation) - Eout(Energy Removal) (21)

Temperature rise via Cp Dielectric Heating (eq.(4)) Convection (eq.(2))

(equation (A1)) Radiation (eq.(3))

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Est,Eg

AirTamb,h Eout

Tsur

AC volts Est,Eg

AirTamb,h Eout

Tsur

AC volts

Figure 11 Schematic of energy balance in Self Heating of piezoelectric disc.

As the sample is operating off resonance, the internal energy generation can be

calculated using the dielectric heating model, equation (4). The thermal energy stored

in any arbitrary block of material can be determined from equation (A1), and

assuming there is no thermal conduction, the energy removal can be determined from

equations, (2) and (3).

Assuming that there are negligible heat losses from the edge of the disc, the

instantaneous temperature rise is given by

p

surmamb

Ct

TTTThtVfT

)(2)(2/tan2 4420 = (22)

where t is the disc thickness. The terms in equation (22) are a mixture of material

parameters, fundamental constants and experimental conditions, where the leastaccessible value is the convection coefficient, h. The convection coefficient is very

dependent on experimental conditions and although it can be estimated it can also

easily be determined experimentally from the cooling curves.

5.1.1 Newtons law of cooling

Newtons law of cooling simply states that the rate of change of the temperature is

proportional to the difference between its own temperature and ambient, where the

proportionality constant is the heat transfer coefficient, h.

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)( oTThdt

dT= (23)

Solutions to this equation take the form

(24)htambamb eTTTtT+= )()( 0

This simple equation can be used to model many thermal problems, such as cooling

cups of coffee, time of death calculations, and also thermal behaviour of piezoelectric

devices. Several workers [35], [36], [37] have used this model to fit to experimental

data of surface temperature changes of self-heated piezoelectric devices, and although

the heat transfer coefficients obtained are dependent on specific device geometry andboundary conditions, it does permit limited predictive capabilities.

20

40

60

80

100

0 20 40 60 80 100Time (seconds)

SurfaceTem

peratureC

Fit h=33

Experimental Points

Figure 12 Cooling curve of thin piezoelectric disc with experimental fit based onNewtons law of cooling.

A value of 33W/m2K for the heat transfer coefficient gives a good fit to the

experimental cooling curve, figure 12. Strictly speaking, this constant applies for the

cooling curve, and it is possible that it may not necessarily be equivalent to the

heating process. Zheng et al [35] has attributed variations in heat transfer coefficients

in active piezoelectric devices, to the increased convection due to the vibrating

surface.

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5.1.2 Model assumptions

The model described by equation (22) has a number of assumptions including;

negligible heat losses from the disc edge, uniform temperature throughout disc,

radiation occurs between a small sample and large enclosure, and most importantly

constant material properties, i.e. temperature invariant.

As an example the model is tested on heating curves when the sample is driven at

5 kHz, with an applied voltage of 93 volts. If the model is used with the typical

material constants, in particular the small signal or low field permittivity and loss

values, this gives us a very poor fit to the experimental results. In the low field

model the values used for permittivity and loss are 2600 and 0.02 respectively,

leading to an overall temperature rise of 1C, figure 13. It is perhaps unrealistic to use

the small signal dielectric constant values, since small signal is generally measured

at fields of around 1V/mm. If the dielectric constants measured at a high field of 94

volts are used, permittivity = 3300 and loss = 0.3, then a stable temperature rise of

18.5C is predicted. The high field model is improved compared with the low

field prediction, however the temperature rise is roughly half that measured.

20

30

40

50

60

70

0 50 100 150 200

Time (seconds)

SurfaceTemperatureC 90volts

93volts

high field model(temperature dependent)high field model

low field model

Figure 13 Experimental results of self heating of thin piezoelectric disc driven at

500Hz, compared with low field and high field models based on equation (22).

The assumption in the model so far has been that all the material constants are

temperature independent, however this may not be a valid assumption. The dielectric

31


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properties, particularly for soft materials are known to be temperature dependent, and

measurements of the voltage and current during the experiment have shown this to be

the case. The temperature dependence can be included in the model by assuming a

linear variation with temperature for the permittivity and loss such that;

)(703300 ambTT+= (25)

)(001.03.0tan ambTT+= (26)

A much improved fit is obtained when the temperature dependence of the dielectric

properties is included in the model, figure 13.

5.1.3 Thermal runaway

As discussed previously, thermal runaway occurs in systems when the internal energy

generation is greater than the energy removal rate and the system becomes unstable,

usually ending in system failure. It is common in chemical processes, charging of

batteries and capacitors [3], many semiconductor devices, as well as piezoelectric

devices. The key feature of this phenomenon is some kind of positive feedback

between the internal energy generation and the increasing temperature.

Initial internal heat generation rate

Increases temperature

Increases heat generation rate

From the modelling of the thin disc it is clear that the temperature dependence of thedielectric properties plays a key part in temperature rise and is a possible mechanism

for this positive feedback.

When the disc is driven with high fields at a higher frequency of 5 kHz, the sample

starts to exhibit thermal runaway, figure 14a. This can be seen more clearly as a

change in the temporal temperature gradient from negative to positive, figure 14b. In

this case at a voltage of approximately 84 volts the behaviour changes from stable

equilibrium temperature to thermal runaway. This behaviour is chaotic and the exact

voltage that this change occurs is highly dependent on slight changes in theexperimental conditions such as the heat removal rate. Figure 14a also shows the

32


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modelled temperature rise (solid lines), based on the temperature dependent high field

model. The fit for the higher voltages, cf. 91 volts, is not as good as at 77 volts and

reflects the chaotic nature of the behaviour near the runaway point. In order to get a

better fit at 91 volts it is necessary to raise the modelled voltage to 100 volts.

a)

0

1

2

3

4

0 20 40 60 80 100

Time (seconds)

Temperaturechange/secondC/s

91 Volts

87.5 Volts

84 Volts

80.5 Volts

77 Volts

b)

Figure 14 Self heating in a thin PZT disc driven at 5 kHz exhibiting thermal

runaway. a) Experimental temperature rise (points) and modelled temperature

rise (solid lines) based on a temperature dependent high field model. b) Rate of

temperature change of disc showing clearly the onset of thermal runaway. Power

is automatically cut off when sample temperature stabilises, or temperature

reaches 100 C, resulting in cooling curves.

20

40

60

80

100

0 20 40 60 80 100

Time (seconds)

SurfaceTemperatureC

91 Volts

87.5 Volts84 Volts

80.5 Volts

77 Volts

Stable

Temperatures

Thermal

Runaway

20

40

60

80

100

0 20 40 60 80 100

Time (seconds)

SurfaceTemperatureC

91 Volts

87.5 Volts84 Volts

80.5 Volts

77 Volts

Stable

Temperatures

Thermal

Runaway

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5.1.4 Conclusion

A simple model based on the energy balance in a volume, where only energy

generation, convection and radiation are considered has been developed that canaccurately predict the temperature rise in thin piezoelectric discs. It has been shown

that this model is vital if the internal heat generation mechanism is temperature

dependent, and it can even extend into prediction of thermal runaway, although here

the chaotic behaviour make accurate predictions difficult. The model is solved

iteratively, and in principle this method can be incorporated into FEA solutions where

complex geometries and thermal conduction can be included.

5.2 Case 2: Resonant drive of simple monolithic ceramicsuspended in air

Rationale: This device was chosen to illustrate the effects of the spatially dependent

heating of a piezoelectric, depending on the strain profile at resonance. The device has

a large surface area to fill the field of view of the thermal camera, and is driven in the

d31 mode, so the direction of electric field is perpendicular to the expected spatial

thermal anomalies

5.2.1 Experimental

A bar of hard PZT 4D material, 70 mm long, 10 mm wide and 2 mm thick is driven at

longitudinal resonance and its overtones. The bar is poled across the width and the

field is also applied in this direction, resulting in expansion along the length. The

sample is driven at the resonance frequency, 24 kHz and the first and second

overtones, 71.2 and 111.1 kHz. The sample is suspended in air on two wooden

cocktail sticks, and thermal images recorded using an infrared camera. The

temperature readings from the camera were corrected for emissivity by placing the

sample on a hot plate and comparing the camera results with those from a type Kthermocouple.

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wire

wire

PZT bar

70mm

support

Poling and Field

Direction

wire

wire

PZT bar

70mm

support

Poling and Field

Direction

22

23

24

25

26

27

28

29

30

0 20 40 60

Distance (mm)

TemperatureC

21

22

23

24

25

0 20 40 60

Distance (mm)

TemperatureC

Thermal image and centreline temperature

profile of first overtone, 71.2kHz.

Thermal image and centreline temperature

profile of second overtone, 111.1kHz.

Figure 15 Thermal images of piezoelectric bar driven at resonance.

The thermal images, figure 15, clearly show different behaviour dependent on the

resonant mode. At 71.2 kHz the first overtone of the length resonance, there are

clearly three hot spots along the length, roughly coincident with the nodes where the

strain is at a maximum. Similarly at the second overtone, 111.1 kHz, there are five hotspots and seven cool areas. The power cables also appear to heat up, being hotter

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where the wire makes contact with the bar. Some of this could be due to contact

resistance as the silver electrode did not make good mechanical contact and was

sometimes pulled off with the wire. Ideally the centre line scans should be perfectly

symmetrical, however there is some evidence that the right hand side of all the images

are nominally hotter. There are several possible reasons for this, including

misalignment of the sample, uneven illumination of the sample and non-uniform pixel

array sensitivity.

5.2.2 Modelling

The modelling of the resonator was performed using a 3D thermal FEA model,

consisting of brick elements with eight nodes, with a single degree of freedom,

temperature, at each node. This element was used for both, steady-state and transient

3-D thermal analysis. The heat loss from the resonator was through a uniform

convection coefficient on all the surfaces, conduction through the wires and supports

and radiation loss was ignored. The convection coefficient was determined by

matching the predicted cooling curves to the measured temperature and a value of 10

W/m2K was used.

The heat generation was modelled by distributing the dissipated power according to

the square of the strain, since both tensile and compressive strains lead to a

temperature rise. In the d31 mode the largest strains occur along the length,x direction,

and the strain in the other directions was assumed to be uniform. Therefore the

dissipated power was distributed according to the following equation

=

L

xnPxP

2sin)( (27)

where P is the power per unit volume, L is the sample length, and n depends on the

resonant mode, odd for resonance and even for antiresonant modes.

Figure 16 shows the measured centreline temperature profile after 50 and 100 seconds

and the transient modelled profiles using n=3 and a power level of 0.2 W/cm3. The

model shows the same peaks and troughs as the real behaviour, and this holds for

different times, however the match is not perfect. The fact that the measured profiles

are hotter towards the right hand side has already been discussed, however the central

point is consistently the hottest part on the measured device, a feature that is not

mirrored in the model behaviour. It is possible that the electrical contacts, which are

coincidentally near the side lobes are the cause of increased heat loss through

conduction.

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21

23

25

27

29

31

0 10 20 30 40 50 60Distance (mm)

TemperatureC

70

Experiment (100 seconds)

Experiment (50 seconds)

FEA model 0.2W/cc (100 seconds)

FEA model 0.2W/cc (50 seconds)

Figure 16 Experimental and modelled temperature profiles for a bar driven at the first

overtone 70.2kHz.

The modelled and measured temperatures for the second overtone, n = 5, are shown in

figure 17. As can be seen, the temperature difference is smaller for this mode, because

the strain variations are closer together. The model still gives a reasonable

representation of real behaviour, but now because of the small temperature rise, the

errors in the measurements are more apparent.

20.5

21.5

22.5

23.5

24.5

25.5

0 10 20 30 40 50 60 70

Distance (mm)

Tem

peratureC

Experiment ( 50 seconds)

Experiment ( 25 seconds)

FEA 0.15W/cc (50 seconds)

FEA 0.15W/cc (25 seconds)

Figure 17 Experimental and modelled temperature profiles for a bar driven at the

second overtone 111.1kHz.

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So far the results of the model have shown the temperature profile of the centreline of

the top surface only. The FEA model is 3D, so it predicts the temperature of the entire

sample, and the surface temperature is shown in figure 18. From this it can be seen

that the predicted temperature is not uniform along the width direction and the

contours around the hot spots are convex, whereas around the cool areas they are

concave. This is due to the thermal edge effects where more heat is lost near the edges

because the material in this region is close to two surfaces. In fact the experimental

behaviour shows the opposite, with the hot spots being concave, and the cool areas

convex. In the model we have assumed that the spatial dependent effects are entirely

in thex, length direction and uniform along the width, y direction. In fact because of

mechanical edge effects, the strain levels near the edge decrease, which would

amplify the thermal edge effects, thus increasing the curve of the contours rather than

reversing them. Although the predicted contours take the wrong form, in practice the

errors in actual temperature levels are less than 0.5C.

Figure 18 Full 3D FEA simulation of temperature map of bar where n=3, first

overtone.

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5.2.3 Conclusion

The case study clearly illustrates the spatial dependence of heat generation in a

piezoelectric device driven at resonance. This behaviour can be modelled byassuming a sine-squared distribution of the power dissipation rather than a uniform

one. Although the effects here are dramatic, this is because the resonator is

mechanically and thermally, well insulated. In many practical situations the resonator

will be attached to another medium, and heat transfer through conduction will

override the spatial dependent effects seen in this case.

5.3 Case 3: Resonant drive of a clamped ceramic (bolted

Langevin transducer)

Rationale: This is an example of an industrial high power transducer and includes all

the problems associated with non-ideal cases. The device geometry is more

complicated, there are many different materials present, and the drive electronics are

proprietary. Because of these complexities, it is no longer viable to solve this using

analytical solutions and finite element methods must be used.

5.3.1 Experiment

Figure 19 Photo of the back to back high power transducer system.

The transducer is a commercial high power device, and in order to drive the system

safely at high powers it is necessary to extract power from the system. This is

achieved by mounting two transducers back to back in the form of a 1:1

piezotransformer, where the output is terminated into a resistive load consisting of

several halogen lights. Figure 19 shows the system, which is held horizontally at the

nodal points by two large steel plates. The transducers have a resonant frequency of

20 kHz, and the active piezoelectric consists of 6 rings of PZT, 5 mm thick and

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50 mm OD, stacked sequentially but wired electrically in parallel. The ceramic is

prestressed by a stainless steel back mass using a high tensile steel bolt which is

attached to the aluminium front horn. The ceramics have silver electrodes, but in

order to make a mechanically robust electrical contact to these a thin metal shim is

placed between each PZT disc. These shims have a small tab to enable the electrical

contact to be made.

Line Scan

150

250

350

450

550

1 51 101 151 201 251

Pixel Number

14-bitPixelValue

Ceramic

shims

End Mass Horn

Line Scan

150

250

350

450

550

1 51 101 151 201 251

Pixel Number

14-bitPixelValue

Ceramic

shims

End Mass Horn

Figure 20 Thermal image of high power transducer driven at an input power of

400 W for 400 seconds, with associated line scan pixel values.

The temperature of the transducer was measured with an Indigo Merlin InSb mid

range infrared camera with a 320 by 256 pixel array. Figure 20 shows a greyscale

thermal image of the device after being driven for 400 seconds at an input power of

400 W. The image shows hot spots associated with the three power cables, and al

Documents

GPG Thermal V6